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Unformatted text preview: CS205 Review Session # 3 Notes Linear Dependence In order to prove that a set of nonzero vectors are linearly independent, we can assume that they are linearly dependent and show that this leads to a contradiction. If a set of vectors V = { v i 6 = | i { 1 , . . . , m }} is linearly dependent, the following three conditions all hold: 1. c i IR (at least two of which are nonzero) such that m i =1 c i v i = 2. v k V such that v k = i 6 = k c i v i = with at least one c i 6 = 0 3. v k V that can be written as a linear combination of a linearly independent subset of V , i.e. v k = ` i =1 c i v i (with some renumbering), where the choice of { c i } is unique . This last formulation gives, in some sense, the minimal linearly dependent subset of V , since the removal of any vector from { v k , v 1 , . . . , v ` } yields a linearly independent set. Deflation Matrices Consider a symmetric n n matrix A where is an eigenvalue of A and q the corresponding (normalized) eigenvector. We define the deflation of A with respect to ( , q ) to be: D ( , q ) = A- qq T We note that this matrix has the following properties:...
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